Having grandparents who live in one of the fanciest upper class cities in Denmark where all the rich people come to holiday in the summer (but turns plain as a pikestaff in winter time), I would say that these names are very commonly heard among the upper class:

It's an interesting mix of new trends, unusual names, and old classics.
I think it depends on the type of wealthy person. Whether they see themselves as trendsetters (Phineas), as the kind whose kids can pull off anything (Lemon), or as traditional old money (Jane).
And the trendsetter names recycle pretty quickly, so that upper class name of 10 years ago is probably top of the charts today, and today's trendsetter name probably sounds weird to most regular people until it gets a rep of being used by successful people, then it'll start sounding less weird. Classics will always be classics and crazy names will always be crazy. It's those middle of the road names where a lot of movement happens.

I don't think it's a surprise that many of the names I associate with class and success are currently popular. We tend to name aspirationally and the lower middle/lower class will usually be a generation behind, having associated names with success and class among their elders/peers they name there children accordingly (thereby incidentally undermining all the prestige that name used to hold).

Additionally the people we see as successful in our day to day lives are usually older (you need time to build that success!), and so successful and classy names skip a generation.

Consider Sophie, an unusual and beautiful classy choice in the late 80s (and name of a good friend of mine).

What is it now? The French version of one of the most popular names going around, and losing all its class distinction to boot. You can bet your behind that in the upper circles it will now be common and the people in the know will abandon it promptly.

Boys tend to not experience such wild trends, but the establishment names are on the rise again (even the ones that were until recently considered absolutely ridiculous), and I've noticed 'old man' names coming back. Not only old man names but names that imply old-maness (all these last names as first names).

I mean no offence in these observations they're just the trends as I see them.

How do you stay ahead of the trend? I don't know...ask your local duchess or something...

*disclaimer, I don't know any duchesses

*when I reflect on this I must admit a grudging admiration for those parents who name ultra yooneekly/or deliberately er, 'low' for the time. They are effectively absenting themselves from the class catchup game played by so many of us (consciously or unconsciously), in many ways Neveah says "No, we're not posh, we're not pretending...so what?". There's something to be said for that.

I think that wealthy names are subtle & classic or near-classic or vintage-eccentric-classic could work, but probably more on the subtle side is best. The kids of some rich folks I know- Henry, Hal, Lauren (f), Anna, Anya, Tyson, Dane, Catherine..these are people who are well off from their families. Some wealthy people who are first gen wealthy have kids named- Isabella, Victoria, Georgia, Calvin, Benjamin. I think that surname names aren't in the same category they were in the 80s when Spencer & Sloan were the ultimate rich-kid names. Of course rich kids could still get named Weston & Archer, but I think the trend is more towards Winston & June at the moment. Plain but rare-ish almost-classics.